Although the two do not completely contradict each other, they do seem ambiguous.
This week's Parsha, Eqev, to the rescue!
The scale tipper: Hakkatuv hashlishi:

3. Dvarim Eqev Ch. 10:2-4 "v'echtov al halluchos" where it is clear that HKBH wrote on the 2nd Luchos.
I think this structure is clear. Therefore, in #2, Moshe probably wrote something else or wrote the dibros upon something else, like parchment.